Power Rule

When we differentiate a power of $x$, we bring the power down and reduce it by $1$.

When we integrate, we do the reverse – first increase the power by $1$, then divide by the new power on the outside.

Things don’t change when you have a negative exponent:

This holds true for fractions as well. When you do have fractional powers in a more involved integral, things can become a bit more difficult to spot.

And of course, it holds for irrationals too…

So generally, for any real $n$¹

Power rule crops up everywhere in integration, since it’s a really fundamental integration method. The most important cases are where $n = 0$ and $n = 1$.

This looks really obvious, but when you’ve just started integration, it can be easy to forget what you’re doing when the variables change after a substitution. Always remember that it doesn’t matter what you call your integrating variable – $x$, $t$, $u$, they’re all just labels – the methods remain the same.

For $n = 0$, you’re just integrating a constant, and this gives back your integrating variable:

It probably feels a bit weird at first integrating ‘nothing’. Just remember you’re not integrating $0$, but $x^0 = 1$. If you really want, you can explicitly write the $1$ in:

[!Tip] Integrating $0$ just gives $c$, since any constant differentiates to $0$. For that we’d explicitly write $\int 0 \ dx$.

Ok, are there any real values of $n$ that power rule doesn’t hold for? There is, actually. We reach a strange case with $n = -1$ as the exponent:

If we try applying power rule, we would increment the power to $0$ (giving $x^0 = 1$, a constant) – but more catastrophically, we’d end up dividing by $0$…

Which is definitely illegal.

This is actually the 1 special case where the power rule doesn’t hold. So $n$ can be any real number except $-1$. We’ll look at how to integrate $x^{-1}$ next in Integrating the Reciprocal.